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This article studies the small weight codewords of the functional code C Herm (X), with X a non-singular Hermitian variety of PG(N, q 2 ). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q 2 ) consisting of q + 1 hyperplanes through a common (N − 2)-dimensional space Π, forming a Baer subline in the quotient space of Π. The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C 2 (Q), Q a non-singular quadric [4], and C 2 (X), X a non-singular Hermitian variety [5].

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On the functional codes defined by quadrics and Hermitian varieties

Bartoli

Boeck

Fanali

et al. 2012

Des. Codes Cryptogr.

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In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219-233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729-1739, 2010; Hallez and Storme, Finite Fields Appl 16:27-35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes , with a non-singular Hermitian variety in PG(N, q (2)). The codewords of this code are defined by evaluating the points of in the quadratic polynomials defined over . We now present the similar results for the functional code . The codewords of this code are defined by evaluating the points of a non-singular quadric in PG(N, q (2)) in the polynomials defining the Hermitian varieties of PG(N, q (2))

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A study of intersections of quadrics having applications on the small weight codewords of the functional codes C2(Q),

Cited by 9 publications

References 19 publications

Curves in a hyperbolic quadric surface with a large number of $${\mathbb{F}_{q}}$$ -points

Curves in a hyperbolic quadric surface with a large number of $${\mathbb{F}_{q}}$$ -points

The small weight codewords of the functional codes associated to non-singular Hermitian varieties

On the functional codes defined by quadrics and Hermitian varieties

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